Proposition 19

Let $\varphi$ be an injective homomorphism of a field $K$ into a ring $A$. For every left vector space $E$ over $K$, the canonical mapping $\phi : E \to \varphi^*(E) = A \otimes_K E$ is injective. Moreover, for every vector subspace $E'$ of $E$, $\varphi^*(E') = A \otimes_K E'$ is canonically identified with a direct factor sub-$A$-module of $A \otimes_K E$ and, with this identification,

The first assertion is a special case of § 5, no. 1, Proposition 4; the second is a special case of no. 7, Proposition 14; finally, to show (35), it suffices to take in $A$ (considered as a right $K$-module) a basis $(a_\lambda)_{\lambda \in L}$ such that $a_{\lambda_0} = 1$ for some index $\lambda_0$ (no. 1, Theorem 2); the elements of $A \otimes_K E$ can be written uniquely as $\sum \lambda a_\lambda \otimes x_\lambda$ with $x_\lambda \in E$ and, for such an element to belong to $A \otimes_K E'$, it is necessary and sufficient that $x_\lambda \in E'$ for all $\lambda$. On the other hand, the elements of $\phi(E)$ are those for which $x_\lambda = 0$ for $\lambda \neq \lambda_0$; for an element $\sum \lambda a_\lambda \otimes x_\lambda$ to belong to $(A \otimes_K E') \cap \phi(E)$, it is necessary and sufficient that $x_\lambda = 0$ for $\lambda \neq \lambda_0$ and $x_{\lambda_0} \in E'$, whence the conclusion.